Reading a Derivative Graph
A graph of \(f'\) is a graph of the derivative values, not a graph of the function \(f\). Its vertical coordinate at \(x\) tells you the slope of \(f\) at that input. If the derivative graph lies above the horizontal axis, those slopes are positive and \(f\) is increasing there. If it lies below the axis, the slopes are negative and \(f\) is decreasing.
Zeros of \(f'\) deserve attention, but a zero alone does not tell you that \(f\) has a local maximum or minimum. What matters is whether the derivative changes sign as \(x\) passes the zero. A graph that crosses the axis changes sign; one that touches the axis and turns back may keep the same sign.
From Derivative Sign to Function Behavior
Let \(f\) be continuous on an interval and differentiable in its interior. The Mean Value Theorem says that for any two points \(a<b\) in that interval, there is a point \(c\) between them such that
Since \(b-a>0\), if \(f'(x)>0\) throughout the interval, then \(f(b)-f(a)>0\) whenever \(a<b\). Thus \(f\) is strictly increasing. If \(f'(x)<0\) throughout the interval, then \(f(b)-f(a)<0\), so \(f\) is strictly decreasing. This explains why the position of the derivative graph relative to the axis determines whether the function rises or falls.
At a point where the derivative graph meets the axis, compare its signs immediately to the left and right. The sign pattern determines whether the function changes direction:
| Sign of \(f'\) before and after \(c\) | Behavior of \(f\) near \(c\) |
|---|---|
| Positive, then negative | Increasing, then decreasing: local maximum |
| Negative, then positive | Decreasing, then increasing: local minimum |
| Positive, then positive | Increasing on both sides: no turn indicated |
| Negative, then negative | Decreasing on both sides: no turn indicated |
For the first two patterns, the sign information gives more than a visual clue. For example, suppose \(f'(x)>0\) just to the left of \(c\), \(f'(x)<0\) just to the right, and \(f\) is continuous at \(c\). On the left, the function increases toward \(c\); on the right, it decreases away from \(c\). Therefore nearby function values are no greater than \(f(c)\), and \(c\) is a local maximum. Reversing the signs gives a local minimum. These conclusions use the behavior on both sides, not merely the fact that \(f'(c)=0\).
Worked Example: A Derivative Graph That Touches the Axis
Worked Example: A Derivative Graph That Touches the Axis
Suppose the function is
First differentiate, showing the terms separately:
The derivative graph meets the horizontal axis at \(x=0\) and \(x=2\). The factor \(x\) changes sign at \(0\), while the squared factor \((x-2)^2\) is positive on either side of \(2\). To confirm the signs on each interval, use one test value:
Thus \(f\) decreases on \((-\infty,0)\), then increases on \((0,2)\) and continues increasing on \((2,\infty)\). The sign changes from negative to positive at \(0\), so \(f\) has a local minimum there. At \(2\), the derivative is zero but remains positive on both sides. The function does not turn at \(2\); the derivative graph only touches the axis.
As a check, differentiate the stated function again to confirm the derivative used:
The function values at the two derivative zeros are \(f(0)=0\) and
In particular, the stationary point at \(2\) has value \(4/3\), but the unchanged positive derivative sign shows it is not a local extremum.
Worked Example: Two Axis Crossings
Worked Example: Two Axis Crossings
Consider
Its first derivative is
The derivative graph crosses the axis at \(x=-1\) and \(x=1\). Check its sign in each interval using one value:
Therefore \(f\) increases on \((-\infty,-1)\), decreases on \((-1,1)\), and increases on \((1,\infty)\). The change from positive to negative at \(-1\) gives a local maximum. The change from negative to positive at \(1\) gives a local minimum.
Evaluate \(f\) at the two points to locate the extrema on the function graph:
So the local maximum is at \((-1,2)\), and the local minimum is at \((1,-2)\). Check the derivative by differentiating the original function: \(\frac{d}{dx}(x^3-3x)=3x^2-3\), as required. The values \(2\) and \(-2\) come from \(f\), not from the derivative graph; the derivative graph identifies the \(x\)-coordinates and the changes in direction.
When the Derivative Is Not Defined
A local extremum can occur even when \(f'\) is undefined at the point. The derivative graph may have a gap, a sharp break, or no value at such an input. In that situation, compare the derivative signs on the two sides if they are available. A negative-to-positive change still indicates a local minimum when the function itself is defined and continuous at the point.
Worked Example: A Minimum at a Point Without a Derivative
Take \(f(x)=|x|\). For \(x<0\), \(f(x)=-x\), so \(f'(x)=-1\). For \(x>0\), \(f(x)=x\), so \(f'(x)=1\). At \(x=0\), the left-hand slope is \(-1\) and the right-hand slope is \(1\), so the derivative does not exist there.
The derivative graph therefore has a horizontal branch at \(-1\) for \(x<0\) and a branch at \(1\) for \(x>0\), with no derivative value at \(0\). Its sign changes from negative to positive, indicating a local minimum. Directly checking the function confirms this: \(f(0)=|0|=0\), while for every \(x\ne0\), \(f(x)=|x|>0\). Thus \(f(0)\) is smaller than all nearby function values.
As a final check on the derivative branches, differentiating \(-x\) gives \(-1\), and differentiating \(x\) gives \(1\). The derivative is indeed undefined at the joining point, where the two one-sided slopes differ.
A Common Misreading: The Slope of the Derivative Graph
The height of the derivative graph gives \(f'(x)\). The slope of that graph is a different quantity: where it exists, it describes how \(f'\) changes, not whether \(f\) is increasing or decreasing. A derivative graph can slope upward while still lying below the horizontal axis.
For a concrete example, let \(f(x)=x^3/3-x\). Then \(f'(x)=x^2-1\). On \(0<x<1\), the derivative graph slopes upward because its slope is \(2x>0\), but its height is \(x^2-1<0\). Therefore \(f\) is decreasing on \((0,1)\), not increasing.
A Reliable Reading Procedure
When a first-derivative graph is supplied, organize the reading around its zeros, gaps, and sign changes. Do not try to infer the function’s exact height from the derivative graph alone. Use the sign of \(f'\) to describe increase or decrease, then check the signs on both sides of a candidate point before classifying it as an extremum.
Locate where the derivative graph meets the horizontal axis and where it is missing or breaks.
Above the axis means \(f'>0\) and \(f\) increases; below it means \(f'<0\) and \(f\) decreases.
A positive-to-negative change indicates a local maximum; a negative-to-positive change indicates a local minimum. No sign change means no turn is indicated by the first-derivative test.
A point outside the domain cannot be a local extremum of the function. At a point in the domain where the derivative is undefined, use the behavior on both sides and the function’s continuity there.
This method turns a first-derivative graph into a description of the function’s direction and local turning behavior. The zeros matter, but their sign changes carry the decisive information.
Check Your Understanding
Use the sign of the derivative graph in each question; do not confuse its height with its slope.
- If \(f'(x)\) is negative throughout an interval, what is the behavior of \(f\) on that interval?
- The derivative graph crosses the axis from above to below at \(x=c\). What local behavior does this indicate for \(f\), assuming the function is defined and continuous there?
- If \(f'(c)=0\) but \(f'\) is positive on both sides of \(c\), what can you conclude about a turn at \(c\)?
- Can a local minimum occur at a point where \(f'\) is undefined? Give a brief explanation.
- For \(f(x)=x^3/3-x\), is \(f\) increasing or decreasing on \((0,1)\), and how does the height of its derivative graph support your answer?
